A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).
0, 4, 4, 79, 79, 79, 3204, 18829, 331329, 1112579, 1112579, 20643829, 118300079, 850721954, 3292128204, 27706190704, 149776503204, 302364393829, 1065303846954, 8694698378204, 46841671034454, 332943965956329, 332943965956329, 5101315547987579, 28943173458143829
Offset: 0
Keywords
Examples
The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.
Links
- Robert Israel, Table of n, a(n) for n = 0..1424
- G. P. Michon, Introduction to p-adic integers, Numericana.
Crossrefs
Programs
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Maple
R:= [padic:-rootp(x^2+9,5,101)]: R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)): seq(padic:-ratvaluep(R,n),n=0..100); # Robert Israel, Jan 16 2023
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PARI
a(n) = truncate(-sqrt(-9+O(5^n)))
Formula
a(1) = 4; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.
For n > 0, a(n) = 5^n - A327302(n).
Comments